a non-embedding result for thompson's group v · 2013. 8. 12. · a non-embedding result for...
TRANSCRIPT
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
A non-embedding result forThompson’s Group V
Nathan Corwin
University of Nebraska – Lincoln
Groups St Andrews 20137 August 2013
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Overview
Theorem (C. 2013)
Z o Z2 does not embed into Thompson’s Group V .
Give some motivation.
Define wreath product.
Define Thompson’s Group V.
Briefly discuss dynamics in the group.
Briefly discuss the proof of the theorem.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Overview
Theorem (C. 2013)
Z o Z2 does not embed into Thompson’s Group V .
Give some motivation.
Define wreath product.
Define Thompson’s Group V.
Briefly discuss dynamics in the group.
Briefly discuss the proof of the theorem.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Overview
Theorem (C. 2013)
Z o Z2 does not embed into Thompson’s Group V .
Give some motivation.
Define wreath product.
Define Thompson’s Group V.
Briefly discuss dynamics in the group.
Briefly discuss the proof of the theorem.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Overview
Theorem (C. 2013)
Z o Z2 does not embed into Thompson’s Group V .
Give some motivation.
Define wreath product.
Define Thompson’s Group V.
Briefly discuss dynamics in the group.
Briefly discuss the proof of the theorem.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Overview
Theorem (C. 2013)
Z o Z2 does not embed into Thompson’s Group V .
Give some motivation.
Define wreath product.
Define Thompson’s Group V.
Briefly discuss dynamics in the group.
Briefly discuss the proof of the theorem.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Overview
Theorem (C. 2013)
Z o Z2 does not embed into Thompson’s Group V .
Give some motivation.
Define wreath product.
Define Thompson’s Group V.
Briefly discuss dynamics in the group.
Briefly discuss the proof of the theorem.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
C Fand coC F
In the mid-1980’s Muller and Schupp showed that theclass of all of groups that have a context free wordproblem (denoted C F ) is equivalent to the the class ofgroups that are virtually free.
A natural generalization of C F is the class coC F : allgroups for which the coword problem is context free.
This class was first introduced by Holt, Rees, Rover, andThomas in 2006.
They showed that coC Fhas many closure properties.Closed under:
direct products;standard restricted wreath products where the top group isC F ;passing to finitely generated subgroups;passing to finite index over-groups.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
C Fand coC F
In the mid-1980’s Muller and Schupp showed that theclass of all of groups that have a context free wordproblem (denoted C F ) is equivalent to the the class ofgroups that are virtually free.
A natural generalization of C F is the class coC F : allgroups for which the coword problem is context free.
This class was first introduced by Holt, Rees, Rover, andThomas in 2006.
They showed that coC Fhas many closure properties.Closed under:
direct products;standard restricted wreath products where the top group isC F ;passing to finitely generated subgroups;passing to finite index over-groups.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
C Fand coC F
In the mid-1980’s Muller and Schupp showed that theclass of all of groups that have a context free wordproblem (denoted C F ) is equivalent to the the class ofgroups that are virtually free.
A natural generalization of C F is the class coC F : allgroups for which the coword problem is context free.
This class was first introduced by Holt, Rees, Rover, andThomas in 2006.
They showed that coC Fhas many closure properties.Closed under:
direct products;standard restricted wreath products where the top group isC F ;passing to finitely generated subgroups;passing to finite index over-groups.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
C Fand coC F
In the mid-1980’s Muller and Schupp showed that theclass of all of groups that have a context free wordproblem (denoted C F ) is equivalent to the the class ofgroups that are virtually free.
A natural generalization of C F is the class coC F : allgroups for which the coword problem is context free.
This class was first introduced by Holt, Rees, Rover, andThomas in 2006.
They showed that coC Fhas many closure properties.Closed under:
direct products;standard restricted wreath products where the top group isC F ;passing to finitely generated subgroups;passing to finite index over-groups.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
C Fand coC F
In the mid-1980’s Muller and Schupp showed that theclass of all of groups that have a context free wordproblem (denoted C F ) is equivalent to the the class ofgroups that are virtually free.
A natural generalization of C F is the class coC F : allgroups for which the coword problem is context free.
This class was first introduced by Holt, Rees, Rover, andThomas in 2006.
They showed that coC Fhas many closure properties.Closed under:
direct products;standard restricted wreath products where the top group isC F ;passing to finitely generated subgroups;passing to finite index over-groups.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
coC F
Two conjectures from that paper:
1 If C o T is in coC F , then T must be in C F ;
My theorem supports this conjecture.
2 coC F is not closed under free products.
The leading candidate to show the second conjecture is thegroup Z ∗ Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
coC F
Two conjectures from that paper:
1 If C o T is in coC F , then T must be in C F ;
My theorem supports this conjecture.
2 coC F is not closed under free products.
The leading candidate to show the second conjecture is thegroup Z ∗ Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
coC F
Two conjectures from that paper:
1 If C o T is in coC F , then T must be in C F ;
My theorem supports this conjecture.
2 coC F is not closed under free products.
The leading candidate to show the second conjecture is thegroup Z ∗ Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
coC F
Two conjectures from that paper:
1 If C o T is in coC F , then T must be in C F ;
My theorem supports this conjecture.
2 coC F is not closed under free products.
The leading candidate to show the second conjecture is thegroup Z ∗ Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
coC F
Two conjectures from that paper:
1 If C o T is in coC F , then T must be in C F ;
My theorem supports this conjecture.
2 coC F is not closed under free products.
The leading candidate to show the second conjecture is thegroup Z ∗ Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
coC F and V
In 2007 Lehnert and Schweitzer showed that R.Thompson’s group V is in coC F .
This was a surprising result.
It also put into doubt the belief that Z ∗ Z2 is not incoC Fas V contains many copies of Z and Z2 and freeproducts of subgroups are common in V .
In 2009, Bleak and Salazar-Dıaz showed that Z ∗ Z2 doesnot embed into V .
In that paper, they conjectured that Z o Z2 does notembed into V .
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
coC F and V
In 2007 Lehnert and Schweitzer showed that R.Thompson’s group V is in coC F .
This was a surprising result.
It also put into doubt the belief that Z ∗ Z2 is not incoC Fas V contains many copies of Z and Z2 and freeproducts of subgroups are common in V .
In 2009, Bleak and Salazar-Dıaz showed that Z ∗ Z2 doesnot embed into V .
In that paper, they conjectured that Z o Z2 does notembed into V .
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
coC F and V
In 2007 Lehnert and Schweitzer showed that R.Thompson’s group V is in coC F .
This was a surprising result.
It also put into doubt the belief that Z ∗ Z2 is not incoC Fas V contains many copies of Z and Z2 and freeproducts of subgroups are common in V .
In 2009, Bleak and Salazar-Dıaz showed that Z ∗ Z2 doesnot embed into V .
In that paper, they conjectured that Z o Z2 does notembed into V .
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
coC F and V
In 2007 Lehnert and Schweitzer showed that R.Thompson’s group V is in coC F .
This was a surprising result.
It also put into doubt the belief that Z ∗ Z2 is not incoC Fas V contains many copies of Z and Z2 and freeproducts of subgroups are common in V .
In 2009, Bleak and Salazar-Dıaz showed that Z ∗ Z2 doesnot embed into V .
In that paper, they conjectured that Z o Z2 does notembed into V .
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
coC F and V
In 2007 Lehnert and Schweitzer showed that R.Thompson’s group V is in coC F .
This was a surprising result.
It also put into doubt the belief that Z ∗ Z2 is not incoC Fas V contains many copies of Z and Z2 and freeproducts of subgroups are common in V .
In 2009, Bleak and Salazar-Dıaz showed that Z ∗ Z2 doesnot embed into V .
In that paper, they conjectured that Z o Z2 does notembed into V .
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Other motivationStructure of the R. Thompson’s groups
Richard Thompson discovered F < T < V in 1965.
In 1999, Guba and Sapir showed that Z oZ embeds into F ,and thus embeds into T and V as well.
In 2008, Bleak showed that Z o Z2 does not embed into F .
In 2009, Bleak, Kassabov, and Matucci showed Z oZ2 doesnot embed into T .
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Other motivationStructure of the R. Thompson’s groups
Richard Thompson discovered F < T < V in 1965.
In 1999, Guba and Sapir showed that Z oZ embeds into F ,and thus embeds into T and V as well.
In 2008, Bleak showed that Z o Z2 does not embed into F .
In 2009, Bleak, Kassabov, and Matucci showed Z oZ2 doesnot embed into T .
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Other motivationStructure of the R. Thompson’s groups
Richard Thompson discovered F < T < V in 1965.
In 1999, Guba and Sapir showed that Z oZ embeds into F ,and thus embeds into T and V as well.
In 2008, Bleak showed that Z o Z2 does not embed into F .
In 2009, Bleak, Kassabov, and Matucci showed Z oZ2 doesnot embed into T .
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Other motivationStructure of the R. Thompson’s groups
Richard Thompson discovered F < T < V in 1965.
In 1999, Guba and Sapir showed that Z oZ embeds into F ,and thus embeds into T and V as well.
In 2008, Bleak showed that Z o Z2 does not embed into F .
In 2009, Bleak, Kassabov, and Matucci showed Z oZ2 doesnot embed into T .
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Some notation
Suppose that a, b ∈ G and G acts on a set X.
We will use right actions.We write (x)a or just xa instead of a(x).
Conjugation: ab = b−1ab.
Commutator: [a, b] = a−1b−1ab = a−1ab = (b−1)ab.
Support of a function (element):Supp (a) = {x ∈ X|xa 6= x}.Note, this differs slightly from the standard analysisdefinition.
Fact: Supp (ab) = Supp (a)b.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Some notation
Suppose that a, b ∈ G and G acts on a set X.
We will use right actions.We write (x)a or just xa instead of a(x).
Conjugation: ab = b−1ab.
Commutator: [a, b] = a−1b−1ab = a−1ab = (b−1)ab.
Support of a function (element):Supp (a) = {x ∈ X|xa 6= x}.Note, this differs slightly from the standard analysisdefinition.
Fact: Supp (ab) = Supp (a)b.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Some notation
Suppose that a, b ∈ G and G acts on a set X.
We will use right actions.We write (x)a or just xa instead of a(x).
Conjugation: ab = b−1ab.
Commutator: [a, b] = a−1b−1ab = a−1ab = (b−1)ab.
Support of a function (element):Supp (a) = {x ∈ X|xa 6= x}.Note, this differs slightly from the standard analysisdefinition.
Fact: Supp (ab) = Supp (a)b.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Some notation
Suppose that a, b ∈ G and G acts on a set X.
We will use right actions.We write (x)a or just xa instead of a(x).
Conjugation: ab = b−1ab.
Commutator: [a, b] = a−1b−1ab
= a−1ab = (b−1)ab.
Support of a function (element):Supp (a) = {x ∈ X|xa 6= x}.Note, this differs slightly from the standard analysisdefinition.
Fact: Supp (ab) = Supp (a)b.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Some notation
Suppose that a, b ∈ G and G acts on a set X.
We will use right actions.We write (x)a or just xa instead of a(x).
Conjugation: ab = b−1ab.
Commutator: [a, b] = a−1b−1ab = a−1ab = (b−1)ab.
Support of a function (element):Supp (a) = {x ∈ X|xa 6= x}.Note, this differs slightly from the standard analysisdefinition.
Fact: Supp (ab) = Supp (a)b.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Some notation
Suppose that a, b ∈ G and G acts on a set X.
We will use right actions.We write (x)a or just xa instead of a(x).
Conjugation: ab = b−1ab.
Commutator: [a, b] = a−1b−1ab = a−1ab = (b−1)ab.
Support of a function (element):Supp (a) = {x ∈ X|xa 6= x}.Note, this differs slightly from the standard analysisdefinition.
Fact: Supp (ab) = Supp (a)b.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Some notation
Suppose that a, b ∈ G and G acts on a set X.
We will use right actions.We write (x)a or just xa instead of a(x).
Conjugation: ab = b−1ab.
Commutator: [a, b] = a−1b−1ab = a−1ab = (b−1)ab.
Support of a function (element):Supp (a) = {x ∈ X|xa 6= x}.Note, this differs slightly from the standard analysisdefinition.
Fact: Supp (ab) = Supp (a)b.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Wreath Products
Let A and T be groups.
Set B = ⊕t∈TA.
Then, the Wreath Product of A and T is A o T = B o T(where the semi-direct product action of T on B is rightmultiplication on the index in the direct product).
We say T is the top group, A is the bottom group, and B iscalled the base group.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Wreath Products
Let A and T be groups.
Set B = ⊕t∈TA.
Then, the Wreath Product of A and T is A o T = B o T(where the semi-direct product action of T on B is rightmultiplication on the index in the direct product).
We say T is the top group, A is the bottom group, and B iscalled the base group.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Wreath Products
Let A and T be groups.
Set B = ⊕t∈TA.
Then, the Wreath Product of A and T is A o T = B o T(where the semi-direct product action of T on B is rightmultiplication on the index in the direct product).
We say T is the top group, A is the bottom group, and B iscalled the base group.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
First look at R. Thompson’s group V
V is finitely presented.
Standard presentation has 4 generators, and 13 relations.
The generators of the standard presentation areA,B,C, π0.
The relations:
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
First look at R. Thompson’s group V
V is finitely presented.
Standard presentation has 4 generators,
and 13 relations.
The generators of the standard presentation areA,B,C, π0.
The relations:
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
First look at R. Thompson’s group V
V is finitely presented.
Standard presentation has 4 generators, and 13 relations.
The generators of the standard presentation areA,B,C, π0.
The relations:
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
First look at R. Thompson’s group V
V is finitely presented.
Standard presentation has 4 generators, and 13 relations.
The generators of the standard presentation areA,B,C, π0.
The relations:
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
First look at R. Thompson’s group V
V is finitely presented.
Standard presentation has 4 generators, and 13 relations.
The generators of the standard presentation areA,B,C, π0.
The relations:
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
First look at R. Thompson’s group V
[AB−1, A−1BA] = 1;
[AB−1, A−2BA2] = 1;
C = BA−1CB;
A−1CBA−1BA = BA−2CB2;
CA = (A−1CB)2;
C3 = 1;
((A−1CB)−1π0A−1CB)2 = 1;
[(A−1CB)−1π0A−1CB,A−2(A−1CB)−1π0A
−1CBA2] = 1;
(A−1(A−1CB)−1π0A−1CBA(A−1CB)−1π0A
−1CB)3 = 1;
[A−2BA2, (A−1CB)−1π0A−1CB] = 1;
(A−1CB)−1π0A−1CBA−1BA =
BA−1(A−1CB)−1π0A−1CBA(A−1CB)−1π0A
−1CB;
A−1(A−1CB)−1π0A−1CBAB =
BA−2(A−1CB)−1π0A−1CBA2;
(A−1CB)−1π0A−1CBA−2CB2 =
A−2CB2A−1(A−1CB)−1π0A−1CBA;
((A−1CB)−1π0A−1CBA−1CB)3 = 1〉
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Second look at R. Thompson’s group V
Let T be the infinite, rooted, directed, binary tree.
Note that the limit space is the Cantor set.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Second look at R. Thompson’s group V
Let T be the infinite, rooted, directed, binary tree.
Note that the limit space is the Cantor set.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
An element of V
Let D and R be two finite connected rooted subgraphs of T(with the same root as T ) both with n leaves for somearbitrary n.Let σ ∈ Sn.Then u = (D,R, σ) is a representative of an element of V .(Tree pair representative)
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
An element of V
Let D and R be two finite connected rooted subgraphs of T(with the same root as T ) both with n leaves for somearbitrary n.
Let σ ∈ Sn.Then u = (D,R, σ) is a representative of an element of V .(Tree pair representative)
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
An element of V
Let D and R be two finite connected rooted subgraphs of T(with the same root as T ) both with n leaves for somearbitrary n.Let σ ∈ Sn.
Then u = (D,R, σ) is a representative of an element of V .(Tree pair representative)
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
An element of V
Let D and R be two finite connected rooted subgraphs of T(with the same root as T ) both with n leaves for somearbitrary n.Let σ ∈ Sn.Then u = (D,R, σ) is a representative of an element of V .(Tree pair representative)
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Example of an element U in V
(0101100 . . . )U
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Example of an element U in V
(0101100 . . . )U
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Example of an element U in V
(0101100 . . . )U
= 101100 . . .
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Example of an element U in V
(0101100 . . . )U = 101100 . . .
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Not unique representation
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Not unique representation
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Not unique representation
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
A simplifying assumption for this talk
I will assume that every element of V ( I discuss ) has nonon-trivial orbits when it acts on the Cantor set.
This is not a big assumption (for me ) as for any v ∈ Vthere is an n ∈ N such that vn has this condition.
This assumption is not needed to understand the dynamicsof V , but makes things simpler to explain.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
A simplifying assumption for this talk
I will assume that every element of V ( I discuss ) has nonon-trivial orbits when it acts on the Cantor set.
This is not a big assumption (for me ) as for any v ∈ Vthere is an n ∈ N such that vn has this condition.
This assumption is not needed to understand the dynamicsof V , but makes things simpler to explain.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
A simplifying assumption for this talk
I will assume that every element of V ( I discuss ) has nonon-trivial orbits when it acts on the Cantor set.
This is not a big assumption (for me ) as for any v ∈ Vthere is an n ∈ N such that vn has this condition.
This assumption is not needed to understand the dynamicsof V , but makes things simpler to explain.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Revealing Pairs
Consider the common tree C = D ∩R.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Revealing Pairs
Consider the common tree C = D ∩R.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Revealing Pairs
Consider the common tree C = D ∩R.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Revealing Pairs and Important points
A revealing pair is a particularly tree pair of an element ofV .
Each element of V has a revealing pair.
Each connected component of D \ C has a unique fixedpoint. We will call it a repelling fixed point.
Each connected component of R \ C has a unique fixedpoint. We will call it an attracting fixed point.
The set of the attracting and repelling fixed points are theset of important points for u. This set is denoted by I(u).
Fact: Supp (a) = Supp (a) ∪ I(a).
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Revealing Pairs and Important points
A revealing pair is a particularly tree pair of an element ofV .
Each element of V has a revealing pair.
Each connected component of D \ C has a unique fixedpoint. We will call it a repelling fixed point.
Each connected component of R \ C has a unique fixedpoint. We will call it an attracting fixed point.
The set of the attracting and repelling fixed points are theset of important points for u. This set is denoted by I(u).
Fact: Supp (a) = Supp (a) ∪ I(a).
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Revealing Pairs and Important points
A revealing pair is a particularly tree pair of an element ofV .
Each element of V has a revealing pair.
Each connected component of D \ C has a unique fixedpoint. We will call it a repelling fixed point.
Each connected component of R \ C has a unique fixedpoint. We will call it an attracting fixed point.
The set of the attracting and repelling fixed points are theset of important points for u. This set is denoted by I(u).
Fact: Supp (a) = Supp (a) ∪ I(a).
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Revealing Pairs and Important points
A revealing pair is a particularly tree pair of an element ofV .
Each element of V has a revealing pair.
Each connected component of D \ C has a unique fixedpoint. We will call it a repelling fixed point.
Each connected component of R \ C has a unique fixedpoint. We will call it an attracting fixed point.
The set of the attracting and repelling fixed points are theset of important points for u. This set is denoted by I(u).
Fact: Supp (a) = Supp (a) ∪ I(a).
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Revealing Pairs and Important points
A revealing pair is a particularly tree pair of an element ofV .
Each element of V has a revealing pair.
Each connected component of D \ C has a unique fixedpoint. We will call it a repelling fixed point.
Each connected component of R \ C has a unique fixedpoint. We will call it an attracting fixed point.
The set of the attracting and repelling fixed points are theset of important points for u. This set is denoted by I(u).
Fact: Supp (a) = Supp (a) ∪ I(a).
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Revealing Pairs and Important points
A revealing pair is a particularly tree pair of an element ofV .
Each element of V has a revealing pair.
Each connected component of D \ C has a unique fixedpoint. We will call it a repelling fixed point.
Each connected component of R \ C has a unique fixedpoint. We will call it an attracting fixed point.
The set of the attracting and repelling fixed points are theset of important points for u. This set is denoted by I(u).
Fact: Supp (a) = Supp (a) ∪ I(a).
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Flow graph
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Flow graph
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Flow graph
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Flow graph
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Flow graph
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Components of support
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
A lemma
Lemma (Bleak, Salazar-Dıaz, 2009)
Suppose g, h ∈ V , each with no non-trivial periodic orbits. For(i) and (ii), suppose further that g and h commute. Then:
i. I(g) ∩ I(h) = I(g) ∩ Supp (h) = I(h) ∩ Supp (g);
ii. If X and Y are components of support of g and hrespectively, then X = Y or X ∩ Y = ∅;
iii. Suppose g and h have a common component of support X,and on X the actions of g and h commute. Then, thereare non-trivial powers m and n such that gm = hn over X.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof (outline) of main result
Recall
Theorem (C. 2013)
Z o Z2 does not embed into Thompson’s Group V .
Proof:
Step 1: Suppose there is an injection φ : Z o Z2 → V
Step 2: Clean up injection
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof (outline) of main result
Recall
Theorem (C. 2013)
Z o Z2 does not embed into Thompson’s Group V .
Proof:
Step 1: Suppose there is an injection φ : Z o Z2 → V
Step 2: Clean up injection
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof (outline) of main result
Recall
Theorem (C. 2013)
Z o Z2 does not embed into Thompson’s Group V .
Proof:
Step 1: Suppose there is an injection φ : Z o Z2 → V
Step 2: Clean up injection
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 2: Clean up injection
Let s′ and t′ be the images of the generators of the Z2.
Raise s′ and t′ to powers to obtain s and t with nonon-trivial finite orbits.
Fix an element γ′0 in the bottom group with no non-trivialfinite orbits.
Repeatedly apply two technical lemmas of Bleak andSalazar-Dıaz to eventually replace γ′0 with γ0.
γ0 will:
be a non-trivial element of the base with no non-trivialfinite orbits;have support disjoint from a neighborhood of theimportant points of s and t.
We have 〈s, t, γ0〉 ∼= Z o Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 2: Clean up injection
Let s′ and t′ be the images of the generators of the Z2.
Raise s′ and t′ to powers to obtain s and t with nonon-trivial finite orbits.
Fix an element γ′0 in the bottom group with no non-trivialfinite orbits.
Repeatedly apply two technical lemmas of Bleak andSalazar-Dıaz to eventually replace γ′0 with γ0.
γ0 will:
be a non-trivial element of the base with no non-trivialfinite orbits;have support disjoint from a neighborhood of theimportant points of s and t.
We have 〈s, t, γ0〉 ∼= Z o Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 2: Clean up injection
Let s′ and t′ be the images of the generators of the Z2.
Raise s′ and t′ to powers to obtain s and t with nonon-trivial finite orbits.
Fix an element γ′0 in the bottom group with no non-trivialfinite orbits.
Repeatedly apply two technical lemmas of Bleak andSalazar-Dıaz to eventually replace γ′0 with γ0.
γ0 will:
be a non-trivial element of the base with no non-trivialfinite orbits;have support disjoint from a neighborhood of theimportant points of s and t.
We have 〈s, t, γ0〉 ∼= Z o Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 2: Clean up injection
Let s′ and t′ be the images of the generators of the Z2.
Raise s′ and t′ to powers to obtain s and t with nonon-trivial finite orbits.
Fix an element γ′0 in the bottom group with no non-trivialfinite orbits.
Repeatedly apply two technical lemmas of Bleak andSalazar-Dıaz to eventually replace γ′0 with γ0.
γ0 will:
be a non-trivial element of the base with no non-trivialfinite orbits;have support disjoint from a neighborhood of theimportant points of s and t.
We have 〈s, t, γ0〉 ∼= Z o Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 2: Clean up injection
Let s′ and t′ be the images of the generators of the Z2.
Raise s′ and t′ to powers to obtain s and t with nonon-trivial finite orbits.
Fix an element γ′0 in the bottom group with no non-trivialfinite orbits.
Repeatedly apply two technical lemmas of Bleak andSalazar-Dıaz to eventually replace γ′0 with γ0.
γ0 will:
be a non-trivial element of the base with no non-trivialfinite orbits;have support disjoint from a neighborhood of theimportant points of s and t.
We have 〈s, t, γ0〉 ∼= Z o Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 2: Clean up injection
Let s′ and t′ be the images of the generators of the Z2.
Raise s′ and t′ to powers to obtain s and t with nonon-trivial finite orbits.
Fix an element γ′0 in the bottom group with no non-trivialfinite orbits.
Repeatedly apply two technical lemmas of Bleak andSalazar-Dıaz to eventually replace γ′0 with γ0.
γ0 will:
be a non-trivial element of the base with no non-trivialfinite orbits;have support disjoint from a neighborhood of theimportant points of s and t.
We have 〈s, t, γ0〉 ∼= Z o Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continued
Proof:
Step 1 Suppose there is an injection φ : Z o Z2 → V
Step 2: Clean up injection
Step 3: Make sequence of γi’s
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continued
Proof:
Step 1 Suppose there is an injection φ : Z o Z2 → V
Step 2: Clean up injection
Step 3: Make sequence of γi’s
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 3: Make sequence of γi’s
Consider the subgroup 〈s, r〉. It has components ofsupport X1, . . . , Xk.
On X1, s and r commute, so there are integers r, q 6= 0such that u = srtq is trivial on X1.
Thus, there is a power p such thatSupp (u) ∩ Supp (γ0) ∩ Supp (γ0)up = ∅. Set w = up.
Define γ1 = [γ0, w]. This element is nontrivial and ofinfinite order and will have no important points in X1.
One can show that 〈s, t, γ1〉 ∼= Z o Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 3: Make sequence of γi’s
Consider the subgroup 〈s, r〉. It has components ofsupport X1, . . . , Xk.
On X1, s and r commute, so there are integers r, q 6= 0such that u = srtq is trivial on X1.
Thus, there is a power p such thatSupp (u) ∩ Supp (γ0) ∩ Supp (γ0)up = ∅. Set w = up.
Define γ1 = [γ0, w]. This element is nontrivial and ofinfinite order and will have no important points in X1.
One can show that 〈s, t, γ1〉 ∼= Z o Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 3: Make sequence of γi’s
Consider the subgroup 〈s, r〉. It has components ofsupport X1, . . . , Xk.
On X1, s and r commute, so there are integers r, q 6= 0such that u = srtq is trivial on X1.
Thus, there is a power p such thatSupp (u) ∩ Supp (γ0) ∩ Supp (γ0)up = ∅. Set w = up.
Define γ1 = [γ0, w]. This element is nontrivial and ofinfinite order and will have no important points in X1.
One can show that 〈s, t, γ1〉 ∼= Z o Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 3: Make sequence of γi’s
Consider the subgroup 〈s, r〉. It has components ofsupport X1, . . . , Xk.
On X1, s and r commute, so there are integers r, q 6= 0such that u = srtq is trivial on X1.
Thus, there is a power p such thatSupp (u) ∩ Supp (γ0) ∩ Supp (γ0)up = ∅. Set w = up.
Define γ1 = [γ0, w]. This element is nontrivial and ofinfinite order and will have no important points in X1.
One can show that 〈s, t, γ1〉 ∼= Z o Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 3: Make sequence of γi’s
Consider the subgroup 〈s, r〉. It has components ofsupport X1, . . . , Xk.
On X1, s and r commute, so there are integers r, q 6= 0such that u = srtq is trivial on X1.
Thus, there is a power p such thatSupp (u) ∩ Supp (γ0) ∩ Supp (γ0)up = ∅. Set w = up.
Define γ1 = [γ0, w]. This element is nontrivial and ofinfinite order and will have no important points in X1.
One can show that 〈s, t, γ1〉 ∼= Z o Z2.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 3: Make sequence of γi’s
Recursively repeat this process: given a γi−1, we can makea γi.
Each time, we have γi is nontrivial and of infinite order.Further, 〈s, t, γi〉 ∼= Z o Z2.
These elements were made so that γi has no importantpoints in Xi.
One can show that γi has no important points in Xj forj < i.
In particular, γk has no important points at all, thus it istrivial.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 3: Make sequence of γi’s
Recursively repeat this process: given a γi−1, we can makea γi.
Each time, we have γi is nontrivial and of infinite order.Further, 〈s, t, γi〉 ∼= Z o Z2.
These elements were made so that γi has no importantpoints in Xi.
One can show that γi has no important points in Xj forj < i.
In particular, γk has no important points at all, thus it istrivial.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 3: Make sequence of γi’s
Recursively repeat this process: given a γi−1, we can makea γi.
Each time, we have γi is nontrivial and of infinite order.Further, 〈s, t, γi〉 ∼= Z o Z2.
These elements were made so that γi has no importantpoints in Xi.
One can show that γi has no important points in Xj forj < i.
In particular, γk has no important points at all, thus it istrivial.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 3: Make sequence of γi’s
Recursively repeat this process: given a γi−1, we can makea γi.
Each time, we have γi is nontrivial and of infinite order.Further, 〈s, t, γi〉 ∼= Z o Z2.
These elements were made so that γi has no importantpoints in Xi.
One can show that γi has no important points in Xj forj < i.
In particular, γk has no important points at all, thus it istrivial.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Proof sketch continuedStep 3: Make sequence of γi’s
Recursively repeat this process: given a γi−1, we can makea γi.
Each time, we have γi is nontrivial and of infinite order.Further, 〈s, t, γi〉 ∼= Z o Z2.
These elements were made so that γi has no importantpoints in Xi.
One can show that γi has no important points in Xj forj < i.
In particular, γk has no important points at all, thus it istrivial.
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Finish proof sketch
Proof:
Step 1: Suppose there is an injection φ : Z o Z2 → V
Step 2: Clean up injection
Step 3: Make sequence of γi’s
Step 4: Show that γk is a nontrivial element of infiniteorder that is also trivial
Step 5: Notice that is a contradiction
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Finish proof sketch
Proof:
Step 1: Suppose there is an injection φ : Z o Z2 → V
Step 2: Clean up injection
Step 3: Make sequence of γi’s
Step 4: Show that γk is a nontrivial element of infiniteorder that is also trivial
Step 5: Notice that is a contradiction
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Finish proof sketch
Proof:
Step 1: Suppose there is an injection φ : Z o Z2 → V
Step 2: Clean up injection
Step 3: Make sequence of γi’s
Step 4: Show that γk is a nontrivial element of infiniteorder that is also trivial
Step 5: Notice that is a contradiction
A non-embeddingresult for
Thompson’sGroup V
NathanCorwin
Introduction
coCFgroups
WreathProducts
Thompson’sGroup V
Dynamics ofV
Proof of MainResult
Thank You
Thank you for your attention.